304 papers
This paper establishes a Galois connection framework for morphisms of neural codes using binary matrices to characterize boolean matrix factorization, define a partial order on codes based on free neurons, and introduce the concept of code defect to analyze covering relations in this poset.
This paper establishes new, improved upper bounds for the classical Ramsey numbers , , and , surpassing the previous limits derived from the standard recursive inequality.
The paper proves that for any fixed maximum degree and any , every sufficiently large oriented graph with minimum semidegree at least contains every oriented tree on vertices with maximum degree at most , a bound that is asymptotically optimal.
This paper extends Turán problems to the spectral domain of digraphs by determining the maximum Laplacian energy and characterizing the extremal digraphs for the class of -free digraphs.
This paper establishes the equivalence of three notions of rank for matrices of multilinear forms, thereby generalizing Flanders' classical result, correcting prior work by Fortin and Reutenauer, resolving a question posed by Lampert, and proving a special case of the conjecture linking analytic and partition ranks of tensors.
This paper proves an asymptotically optimal bound for the concentration function of a sum of independent integer random variables, confirming that the sum's maximum point probability is bounded by that of a corresponding sum of minimal-variance variables when the total variance is sufficiently large, thereby extending the result to separable Hilbert spaces.
This paper proves that for the standard triangle polygon (corresponding to the toric surface ), the spectral transform establishes a birational isomorphism between the Goncharov-Kenyon cluster integrable system and the Beauville integrable system by showing that it intertwines their respective Poisson structures, thereby demonstrating that Beauville integrable systems admit cluster algebra structures.
This paper establishes limit theorems for the number of fixed points in random pattern-avoiding permutations under a biased distribution, revealing a phase transition where the limiting behavior shifts from negative binomial to Rayleigh to normal depending on the bias parameter.
This paper establishes that for any positive integer , there exists a constant such that if a skew circuit in a binary matroid with circumference is contained in a sufficiently large set (where ), then the sum of the sizes of and any other skew circuit is strictly less than $2c - k$.
This paper establishes a symmetric multivariate Elekes-Rónyai theorem providing a lower bound on the size of the image of a polynomial over finite sets unless the polynomial exhibits specific additive or multiplicative degeneracies, while also proving a generalized Erdős-Szemerédi theorem for two polynomials in higher dimensions.
This paper establishes that the tropical rank of a semimodule of tropical rational functions equals its topological dimension, while demonstrating that verifying tropical independence is equivalent to solving a turn-based stochastic mean-payoff game and that computing the tropical rank is NP-hard.
This paper establishes explicit criteria, based on Yavicoli-thickness, -uniformity, and Newhouse thickness conditions, under which compact subsets of (particularly in the plane) are guaranteed to contain similar copies of any linear 3-point configuration or any triangle.
This paper introduces the concept of the Hamiltonian path index as the minimum number of iterations required for a graph's line graph to contain a Hamiltonian path, proving its existence for all graphs and determining its exact value for trees while discussing its properties in graphs with Hamiltonian 2-connected blocks.
This paper defines a compactification of the parameter space for the quantum multiplication of hypertoric varieties, following deConcini and Gaiffi, and demonstrates how to extend this multiplication to the compactified space.
This paper introduces block-separated overpartitions, a constrained family where no two consecutive distinct part-blocks are both overlined, and demonstrates that their enumeration is governed by Fibonacci-type combinatorics, leading to a rich algebraic structure including Euler products and recurrence relations, while sharing the same exponential growth rate as ordinary partitions.
This paper establishes that the homotopy type of the independence complex of a generalized Mycielskian graph is determined by the homotopy types of the independence complexes of the original graph and its Kronecker double cover, and applies this result to compute these types for paths, cycles, and categorical products of complete graphs.
This paper establishes three distinct representations for the flat space wavefunction derived from a graph , proving their correctness by linking the wavefunction to the canonical form of the cosmological polytope and confirming a conjecture regarding its partial fraction decomposition in terms of connected subgraphs.
This paper investigates the homotopy types and connectivity of total cut complexes and their Alexander duals for various graph families, including powers of cycles, complete multipartite graphs, and Cartesian products, thereby resolving specific conjectures regarding these structures.
This paper improves the known bounds for the constant , which represents the minimum proportion of a Sidon subset guaranteed within any -set of size , by establishing that through a reformulated extremal problem and an explicit construction.
This paper establishes that the inversion walk on tournaments exhibits total-variation cutoff at time and characterizes the state space of the -restricted inversion walk as a coset of a subgroup whose codimension depends solely on .